One–Step Hybrid Block Scheme for the Numerical Approximation for Solution of Third Order Initial Value Problems

Abstract

In this research, we have proposed the simulation of linear block algorithm for modeling third order highly stiff problem without reduction to a system of first order ordinary differential equation, to address the weaknesses in reduction method. The method is derived using the linear block method through interpolation and collocation. The basic properties of the block method were recovered and was found to be consistent, convergent and zero-stability. The new block method is been applied to model third order initial value problems of ordinary differential equations without reducing the equations to their equivalent systems of first order ordinary differential equations. The result obtained on the process on some sampled modeled third order linear problems give better approximation than the existing methods which we compared our result with.

On the Simulation of Higher Order Linear Block Algorithm for Modelling Fourth Order Initial Value Problems

“…The following algorithm shows the steps involved in developing the specific block method. LBA follows the concept of considering the general form of the block method while implementing a step-by-step wise mode to obtain the expected block method for solving fourth order ODEs (1.1) [4]. The first algorithm Obtain the block method from the given expression…”

“…These initial approaches include the approach of reducing the higher order ODEs to a system of first order ODEs and then suitable numerical methods such as Simpsons method, Euler's method, Trapezoidal rule, Runge Kutta methods, amongst others are then applied to solve the system (1), [2][3][4][5]. The process of reduction is able to obtain the required solution, but, there is too much rigor involved in the process of reduction, wastage of human effort and also the time constraint.…”

Numerical Application of Higher Order Linear Block Method for Solving Some Fourth Order Initial Value Problems

“…Numerical analysis is the field of mathematics which carries a suitable method for solving difficult problems in mathematics and find out suitable material from accessible results which are not stated in tractable forms. Such problems initiate for the most part, from real world applications of algebra, geometry, calculus, and they include variables which vary [1].…”

“…Someone can reason that, mathematical modeling is an action or procedure that permits a mathematician to adopt the office of a researcher, physicist, chemist, biologist, an ecologist, an economist, a physiologist and so on. Nevertheless, it place of taking research in linear multistep method, a scientist takes a research on mathematical illustrations of the natural processes or physical-world problems (real life problem) [1].…”

“…The system is arranged from first order to higher order problems [2]. Some examiners in literature have treat (1), without following the reduction method because of its setbacks, among others are [2][3][4][5][6][7].…”